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Abstract and Applied Analysis
Volume 2013, Article ID 705126, 13 pages
http://dx.doi.org/10.1155/2013/705126
Research Article

New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 29 May 2013; Revised 17 October 2013; Accepted 22 October 2013

Academic Editor: Chengming Huang

Copyright © 2013 Bin Zheng and Qinghua Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. Z. Li, F. W. Meng, and P. J. Ju, “Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay,” Journal of Mathematical Analysis and Applications, vol. 377, pp. 853–862, 2010. View at Google Scholar
  4. Q. Feng and B. Zheng, “Generalized Gronwall-Bellman-type delay dynamic inequalities on time scales and their applications,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7880–7892, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Q.-H. Ma and J. Pečarić, “The bounds on the solutions of certain two-dimensional delay dynamic systems on time scales,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2158–2163, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. W. S. Wang, “A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation,” Journal of Inequalities and Applications, vol. 2012, article 154, 2012. View at Google Scholar
  7. A. Gallo and A. M. Piccirillo, “About some new generalizations of Bellman-Bihari results for integro-functional inequalities with discontinuous functions and applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. e2276–e2287, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Q.-H. Ma, “Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 233, no. 9, pp. 2170–2180, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. G. Sun, “On retarded integral inequalities and their applications,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 265–275, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. O. Lipovan, “Integral inequalities for retarded Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 349–358, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. P. Agarwal, S. F. Deng, and W. N. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. B. G. Pachpatte, Inequalities for Differential and Integral Equations, vol. 197 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1998. View at MathSciNet
  13. S. H. Saker, “Some nonlinear dynamic inequalities on time scales,” Mathematical Inequalities & Applications, vol. 14, no. 3, pp. 633–645, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Q. Feng, F. Meng, and B. Zheng, “Gronwall-Bellman type nonlinear delay integral inequalities on time scales,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 772–784, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. A. C. Ferreira and D. F. M. Torres, “Generalized retarded integral inequalities,” Applied Mathematics Letters, vol. 22, no. 6, pp. 876–881, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Q. H. Feng, F. W. Meng, and Y. M. Zhang, “Generalized Gronwall-Bellman-type discrete inequalities and their applications,” Journal of Inequalities and Applications, vol. 2011, article 47, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. B. Zheng, Q. H. Feng, F. W. Meng, and Y. M. Zhang, “Some new Gronwall-Bellman type nonlinear dynamic inequalities containing integration on infinite intervals on time scales,” Journal of Inequalities and Applications, vol. 2012, article 201, 2012. View at Google Scholar
  18. S. H. Saker, “Some nonlinear dynamic inequalities on time scales and applications,” Journal of Mathematical Inequalities, vol. 4, no. 4, pp. 561–579, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. Agarwal, M. Bohner, and A. Peterson, “Inequalities on time scales: a survey,” Mathematical Inequalities & Applications, vol. 4, no. 4, pp. 535–557, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. W. S. Wang, “Some retarded nonlinear integral inequalities and their applications in retarded differential equations,” Journal of Inequalities and Applications, vol. 2012, article 75, 2012. View at Google Scholar
  21. B. G. Pachpatte, “Explicit bounds on certain integral inequalities,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 48–61, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. W. N. Li, “Some delay integral inequalities on time scales,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 1929–1936, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y.-H. Kim, “Gronwall, Bellman and Pachpatte type integral inequalities with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. e2641–e2656, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. W. N. Li, M. Han, and F. W. Meng, “Some new delay integral inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 180, no. 1, pp. 191–200, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Q. H. Feng, F. W. Meng, Y. M. Zhang, B. Zheng, and J. C. Zhou, “Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations,” Journal of Inequalities and Applications, vol. 2011, article 29, 2011. View at Google Scholar
  26. W.-S. Cheung and J. L. Ren, “Discrete non-linear inequalities and applications to boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 708–724, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,” Applied Mathematics Letters, vol. 22, no. 3, pp. 378–385, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. B. Zheng, “(G'/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics,” Communications in Theoretical Physics, vol. 58, no. 5, pp. 623–630, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Q. H. Feng, “Exact solutions for fractional differential-difference equations by an extended Riccati Sub-ODE Method,” Communications in Theoretical Physics, vol. 59, pp. 521–527, 2013. View at Google Scholar
  31. S. Zhang and H.-Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. R. Almeida and D. F. M. Torres, “Fractional variational calculus for nondifferentiable functions,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3097–3104, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. Y. Khan, Q. Wu, N. Faraz, A. Yildirim, and M. Madani, “A new fractional analytical approach via a modified Riemann-Liouville derivative,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1340–1346, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. N. Faraz, Y. Khan, H. Jafari, A. Yildirim, and M. Madani, “Fractional variational iteration method via modified RiemannCLiouville derivative,” Journal of King Saud University—Science, vol. 23, pp. 413–417, 2011. View at Google Scholar
  35. Y. Khan, N. Faraz, A. Yildirim, and Q. Wu, “Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2273–2278, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. M. Merdan, “Analytical approximate solutions of fractionel convection-diffusion equation with modified Riemann-Liouville derivative by means of fractional variational iteration method,” Iranian Journal of Science and Technology A, vol. 37, no. 1, pp. 83–92, 2013. View at Google Scholar · View at MathSciNet
  37. S. Guo, L. Mei, and Y. Li, “Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 5909–5917, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. F. C. Jiang and F. W. Meng, “Explicit bounds on some new nonlinear integral inequalities with delay,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 479–486, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. H. P. Ye, J. M. Gao, and Y. S. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet